Numerical Methods in Soil Mechanics 10.PDF

Numerical Methods in Soil Mechanics 10.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.

Anderson, Loren Runar et al "RING STABILITY" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 10-1 Top — Elliptical ring with uniform radial pressure P acting on it Bottom — Free-body-diagram of half of the elliptical ring showing approximate ring compression thrust in the walls due to pressure P ©2000 CRC Press LLC CHAPTER 10 RING STABILITY The performance limit of ring stability is instability Ring instability is a spontaneous deformation that progresses toward inversion (reversal of curvature) At worst, instability is ring collapse Buried pipes can invert only if the ring deflects and the soil slips at the same time Instability of buried pipes is analyzed as a soil-structure interaction The stiffness of the ring resists inversion Soil supports the ring by holding it in a stable (near circular) shape Soil resists inversion of the ring Two basic modes of ring instability are: ring compression, i.e., wall crushing or buckling at yield stress; and ring deformation See Figure 10-1 Each is analyzed separately Instabilities of buried and unburied rings are also analyzed separately Performance limit is wall crushing The ring deformation collapse equation is a function of ring stiffness, EI/r3 Performance limit is inversion Ring stiffness is related to pipe stiffness; i.e., F/∆ = 53.77EI/D3 Pipe stiffness can be measured by a parallel plate test or three-edge-bearing test The ring deformation collapse equation is based on assumptions that the ring is elastic, and that the pipe is restrained longitudinally Longitudinal restraint results in a plane stress analysis The Poisson ratio is not included In a plane strain analysis, longitudinal stress is zero, the Poisson ratio is included, and the pressure at collapse is, Pcr = 3EI/r3(1-ν2), UNBURIED RING COLLAPSE From Chapters and 5, external pressure at collapse of an unburied thin-walled, circular, elastic ring is found from the equations: Pr/A = σf RING COMPRESSION COLLAPSE Pr3/EI = RING DEFORMATION COLLAPSE where P = external pressure at collapse, r = mean circular radius of ring, A = wall area per unit length, t = wall thickness of plain pipe, m = r/t = ring flexibility, = initial ring deflection (ellipse), E = modulus of elasticity, I = moment of inertia of the cross-sectional area of the wall per unit length, σf = yield strength, Pcr = critical pressure on the circular ring The ring compression collapse equation is a function of ring flexibility and yield strength ©2000 CRC Press LLC where Pcr ν EI/r3 = critical pressure, i.e., P at collapse, = Poisson ratio = 0.27 for steel, = ring stiffness The difference in fluid pressures between top and bottom of the pipe is usually ignored, but may be significant For plane stress analysis of critical pressure at collapse of circular, unburied pipes, Pcr = 3EI/r3 (10.1) For plain pipes (not coated, lined, rib stiffened, or corrugated), critical pressure at collapse is, Pcr = E/4m3 (10.2) Moment of Inertia, I In order to evaluate ring stiffness, EI/r3, the moment of inertia, I, must be known For plain pipes, I = t3/12 For corrugated pipes, tables of values for I are found in the manuals provided by manufacturers For steel pipes with lining and coating, consider Figure 10-2 Transformed section of a unit slice of mortar-lined and coated steel pipe wall, transformed into its equivalent section in mortar, for evaluating moment of inertia Shown on the right is the elastic stress distribution Figure 10-3 Effective T-section , comprising stiffener ring and an effective width of pipe wall — often assumed to be 50t in steel pipes ©2000 CRC Press LLC a unit slice of the wall See Figure 10-2 Discounting conservatively the bond between mortar and steel, moment of inertia, I, is the sum of the separate moments of inertia of steel, lining, and coating Because the mortar is the critical material, steel is transformed into its equivalent width, n, in mortar n = Es /Em For the layers, Ic = tc 3/12 Is = nts3/12 Il = tl3/12, and I = Ic + Is + Il For pipes with stiffener rings welded to the pipe, the moment of inertia is found from the effective Ts ection See Figure 10-3 The procedure is described in texts on mechanics of solids For steel pipes, the T-section comprises the stiffener ring and an effective width of pipe wall — in steel usually assumed to be 50t The pipe wall between effective T-sections is ignored in calculating I Elliptical Ring Instability Instability of non-circular, unburied rings is difficult to analyze However, analysis is available from texts on mechanics of solids for one important case — a ring that is initially elliptical with ring deflection d o Vacuum increases ring deflection Stress is c ritical at the spring lines, B, where the ring is subjected to both maximum ring compression stress and maximum flexural stress See Figure 10-1 Vacuum, P, at collapse is found from: P2 - [σf /m + (1+6mdo)Pcr] P + σ f Pcr /m = (10.3a) Equation 10.3a is applied by plotting values of P as a function of m for given values of and for a constant σf For design, a simplification is proposed by Murphy and Langner (1985), P = Pcr /(1+40d) ©2000 CRC Press LLC (10.3b) Example A plain 18-inch high-density polyethylene pipe is out-of-round (elliptical) by five percent It is unburied What is the internal vacuum at collapse? ID DR m σf E Pcr = 16.217 and t = 0.857, = 21 = OD/t = 2m+1, = 10 = r/t, = 3.2 ksi = yield strength', = 110 ksi = modulus of elasticity, = 27.5 psi = E/4m3 from Equation 10.2, = 0.05 = initial ring deflection Substituting into Equation 10.3a, internal vacuum at collapse is, P = 21.5 psi From Equation 10.3b, P = 9.2 psi with ample safety factor included Example Calculate the vacuum at collapse of a mortar-lined and coated steel pipe, for which: r tc ts tl Em Es n σf I d = 25.5 inch (D = 51 inch for steel), = 0.75 inch, Ic = 0.03516 (cr) = 0.175 inch, nIs = 0.00335 = 0.5 inch, Il = 0.01042 = 4(106) psi, = 30(106) psi, = 7.5 = Es /Em , = 10 ksi for mortar (critical), = 0.04893 in 3, = = ring deflection (negligible) From Equation 10.1, Pcr = 3Σ (Em I/r3) Assuming the mean radius of the steel is, rs = 25.5, then rc = 25.9625 and rl = 25.1625 Substituting values, Pcr = 32 psi Example What does Equation 10.3a reduce to if Pcr = P? It is easily shown that all of the terms cancel except 6mdoP, which must be zero The only way this term can be zero is if initial ring deflection is = As expected, for a circular ring, P = Pcr BURIED RING COLLAPSE Stress Analysis of Elliptical Ring Figure 10-1 is a half ring free-body-diagram At B the ring compression stress is, σc = P(OD)(1+d)/2A (10.4) and the ring deformation stress (flexural stress) is, σd = Ec(1/r'x - 1/rx) (10.5) Notation: P = vertical external pressure, OD = outside diameter of the circular ring, = initial ring deflection, d = ring deflection after P is applied, r'x = radius of curvature at B due to initial ring deflection do, rx = radius of curvature at B after vacuum P is applied, E = modulus of elasticity, c = distance from the neutral surface of the wall to the most remote fiber (t/2 for a plain pipe), σd = flexural stress caused by ring deformation Substituting in values of radii of curvature for the ellipse from Chapter 3, the equation for flexural (ring deflection) stress becomes: σd = (Ec/r) 3(d-do) / (1-2d-2do) (10.6) The maximum stress is the sum of Equations 10.4 and 10.6; i.e., σ= σ c + σ f The maximum stress at B in a buried plain pipe is, σ = Pm(1+d) + (E/2m)3(d-do) / (1-2d-2d o) (10.7) where m = r/t = ring flexibility of a plain pipe Equation 10.7 can be solved to find vacuum P at yield stress for brittle (rigid) pipes But yield stress is not failure for plastics or elasto- ©2000 CRC Press LLC plastics (metals), for which wall crushing can occur only after ring compression stress (not flexural stress) reaches yield strength See Chapter Therefore, Equation 10-7 is limited Initial ring deflection, do, depends upon compression of sidefill which requires analysis of pipe-soil interaction Equations 10.4 to 10.7 are based on elastic theory Under some circumstances, plastic theory is justified For plain pipes and corrugated pipes, the plastic moment (at plastic hinging) is 3/2 times the moment at yield stress by elastic theory Ring Deformation Collapse of Buried Pipes For the following analyses, vacuum is negative pressure, p, inside the pipe plus positive external hydrostatic pressure, u Both affect ring collapse Rigid Pipes Because ring deflection of rigid pipes is negligible, rigid pipes are analyzed by ring compression except that vertical pressure on the pipe is P+p Ring compression stress is, σ = (P+p)(OD)/2A (10.8) where σ = ring compression stress in the pipe wall, P = total soil pressure at the top of the pipe, including water pressure, u, p = internal vacuum, OD = outside diameter of the pipe, A = wall area per unit length of pipe Area A is a transformed section if the wall is composite such as concrete reinforced with steel bars For design, the ring compression stress, σ , from Equation 10.8 is equated to the strength of the pipe wall, σf, reduced by a safety factor In the case of a very large diameter pipe, it may be necessary to consider the change in pressure of liquids (both inside and outside) throughout the depth of the pipe For example, if the pipe is empty, but the water table is above the top of the pipe, it may be prudent to apply Equation 10.8 to the bottom of the pipe where total pressure P, acting up on the bottom, is greater than prismatic soil pressure on top by the increase in hydrostatic pressure between the top and bottom Of course, water inside the pipe will negate any increase in external hydrostatic pressure It is noteworthy that internal vacuum and external hydrostatic pressure have little effect on the opening of cracks in rigid pipes The 0.01-inch crack is not a suitable performance limit σf = (P+p)(1+d)OD/2A (10.9) COLLAPSE BY WALL CRUSHING where σf = ring compression stress at yield stress in the pipe wall at B, A = wall area per unit length of pipe, t = wall thickness = A for plain pipes, OD = outside diameter, P = external pressure at top of pipe, p = internal vacuum, d = ∆ D = ring deflection Area, A, is used for transformed composite sections, or ribbed, or ring-stiffened or corrugated Flexible Pipes Collapse of buried flexible pipes is either: wall crushing (ring compression) or inversion (ring deformation) Collapse due to longitudinal bending is not included in this analysis Bending deforms the pipe cross section into an ellipse with the short diameter in the plane of the bend Bending strength is decreased Bending failure is collapse Following are procedures for evaluating the vacuum at which a buried flexible ring collapses If the ring could be held circular, analysis would be simple ring compression — the same as for a rigid pipe But flexible ring analysis anticipates ring deflection, do, before the vacuum is applied Ring deflection depends upon ring stiffness and stiffness of the embedment soil It is assumed that pipes are initially circular and empty, and that coefficient of friction between the pipe and the backfill is zero because of the inevitable breakdown of shearing stresses due to earth tremors and changes in temperature, moisture, and pressures The embedment is assumed to be granular The flexible pipe is often assumed to be thin-walled; i.e., OD = ID = D = mean diameter of the pipe Performance limit is collapse which occurs if the ring either crushes due to ring compression, or inverts due to sidefill soil slip at B See Figure 10-4 ©2000 CRC Press LLC At ring deformation collapse the soil must slip in order for the ring to deflect See Figure 10-4 In the left sketch, the vertical pressure includes soil pressure and vacuum; i.e PA = P+p Before it is buried, the ring is circular, but as backfill is placed, the ring deflects into an ellipse If the ring is flexible, and if shearing stresses between pipe and soil are neligible, vertical and horizontal soil pressures are related as follows: PArA = PBrB = Pr = constant where Pr is the product of pressure and radius of curvature at any point on the circumference of the ring For an ellipse, rA /rB = (1+d)3/(1-d)3 Therefore, PB = PA(1+d)3/(1-d)3 = PAr r (10.10) where rA = mean radius of curvature at the top A, rB = mean radius of curvature at the side B, PA = pressure on the pipe at A, PB = pressure on the pipe at B, d = /D = initial ring deflection, rr = (1+d)3/(1-d)3 = ratio of radii p Figure 10-4 UNSATURATED SOIL — (left) Free-body-diagram of an infinitesimal cube at spring line, B, showing the stresses at incipient soil slip (right) Vertical soil pressure, Po, supported by the pipe due to ring stiffness Figure 10-5 Free-body-diagrams for finding the vertical deflection of point B by means of the Castigliano theorem ©2000 CRC Press LLC But for any pipe stiffness, F/∆ (or equivalent ring stiffness, 53.77 EI/D3), the ring itself is able to support part of the vertical pressure as it deflects That part of the vertical pressure supported by ring stiffness is Po shown in the sketch on the right of Figure 10-4 For a given ring deflection, Po can be calculated by the Castigliano theorem from the freebody-diagram in Figure 10-5 From the sketch on the left, the moment M at point B can be evaluated by noting that the slope at B does not change during deflection θ BA = Knowing M, Castigliano can be applied again using the sketch on the right from which the vertical deflection, YB, is evaluated for the virtual load p due to the forces on the ring quadrant Knowing YB, the ring deflection, d, can be found as a function of EI and Po, and from this relationship, Po can be found as a function of d The result is: PB = (PA+p-Po)rr - p See Figure 10-6 for free-body-diagram and assumptions For a plain pipe, substituting in Po, PB = (PA+uA+p-Ed/m3)rr - p (10.11) Po = 96(EI/D3)d/(1-2d) where PB = PA = p = EI/D3 = F/ = D = r = t = m = d = rr = For small ring deflections, it is conservative to disregard 2d in the denominator; whereupon, µA Po = Ed/m3 = 96(EI/D3)d If soil at B does not have adequate strength, the soil slips, and the ring inverts where Po = Ed/m3 EI/D3 F/∆ I = = = = t D r m d = = = = = vertical pressure on top of the pipe that can be supported by ring stiffness, Po for a plain pipe, ring stiffness = 0.0186 F/∆ , pipe stiffness, moment of inertia of the pipe wall per unit length of pipe = t3/12 for plain pipe, wall thickness for plain pipe, mean diameter, mean radius of the circular pipe, r/t = ring flexibility, ring deflection horizontal pressure of pipe on soil, vertical external soil pressure at A, internal vacuum, ring stiffness, 53.77(EI/D3) = pipe stiffness, mean diameter of the circular ring, mean radius of the circular ring = D/2, thickness of the plain pipe wall, r/t = ring flexibility, ∆ /D = initial ring deflection, (1+d) /(1-d)3 = ratio of vertical and horizontal radii (maximum and minimum radii of the ellipse) = rwh for floods at level h Strength of Soil at Spring Lines Because most embedment is granular, the following is analysis of strength for granular (cohesionless) sidefill See Figure 10-6 The horizontal strength of soil at point B, at soil slip, is soil passive resistance, σx = Kσ y Pressure Against Soil at Spring Lines _where σ = horizontal effective soil stress at B, _x σ y = vertical effective soil stress at B, K = ratio of horizontal to vertical effective stresses at soil slip (ring collapse), K = (1+sinφ)/(1-sinφ), φ = friction angle of the embedment, for which values can be obtained from tests The horizontal pressure of the pipe against the soil at B is reduced by Po; i.e., _ σ y can be evaluated at the spring lines by methods of Chapter ©2000 CRC Press LLC x x Figure 10-6 SATURATED SOIL — Free-body-diagram of an infinitesimal soil cube at B, showing the stresses acting on it at incipient soil slip, and showing the shear planes at soil slip x +u+p Figure 10-7 Pressure diagram for analyzing critical hydrostatic pressure on the bottom of the pipe at inversion of the ring from the bottom rr = ry /rx ©2000 CRC Press LLC Vacuum at Collapse of Buried Pipes The total horizontal soil pressure on the pipe at spring lines B at soil slip is, _ PB = Kσy + uB (10.12) where PB_ = total horizontal pressure on soil at B, Kσ y = horizontal effective soil slip stress at B, uB = hydrostatic pressure in the soil at B When the horizontal pressure PB from Equation 10.11 is equal to PB from Equation 10.12, the soil is on the verge of slipping — instability For plain pipes, including all of the pertinent variables, the equation of equilibrium of sidefill at soil slip is, _ p'(r r-1) = Kσ y + uB - (PA - Ed/m3)rr COLLAPSE BY RING INVERSION (10.13) where p' = vacuum at collapse, rr = (1+d)3/(1-d)3 = ratio of vertical to horizontal radii of elliptical pipe, m = r/t, r = mean radius of the circular pipe, t = wall thickness for plain pipe, d = /D = initial ring deflection — usually due to backfilling, K = (1+sinφ)/(1-sinφ) at passive resistance, φ = friction angle of the embedment, σy = vertical effective soil stress at B, uB = hydrostatic pressure (pore water pressure) at B if a water table is above the pipe, PA = soil and water pressure at A, E = modulus of elasticity of pipe material, I = moment of inertia of the wall cross section per unit length of pipe, E/m = 96EI/D3 where,EI/D3 = ring stiffness, F/∆ = pipe stiffness = 53.77EI/D3 ©2000 CRC Press LLC From Equation 10.13, the vacuum at ring collapse can be calculated E/m can be replaced by 96EI/D3, or 1.7856F/∆ for other-than-plain pipes If the embedment is not cohesionless, as assumed in the above analysis, the same procedure may be used except that the relationship between horizontal and vertical stresses at soil slip must be evaluated for each particular _embedment In the case of ideal _ cohesive soil, σ y - σ x = 2C, where C is the cohesion of the soil See Chapter Below a groundwater table, the hydrostatic pressure on the bottom of the pipe is greater than on top Figure 10-7 shows buoyant pressure on the bottom, γ w(h+H+D) An empty pipe tends to float, but in this analysis, is assumed to be restrained by the effective soil wedge on top Collapse occurs from the bottom for large, empty, flexible pipes with a water table above the pipe Example A thin-wall, welded steel penstock is 51 inches in diameter with a wall thickness of 0.219 inch It is buried in embedment of dry, uncompacted sand to a height of two ft above the top of the pipe Unit weight of the sand is 102 pcf The soil friction angle is 25 o Ring deflection was not controlled during backfilling, so the average initial ring deflection is 8% What is the internal vacuum at collapse? The pertinent data are: D = 2r = 51 inches, t = 0.219 inch = average wall thickness, E = 30(106) psi = modulus of elasticity, I = 875(10-6) in3 = moment of inertia of the wall cross section per inch of length of pipe EI/D3 = 0.198 psi = ring stiffness, d = 0.08 = ∆/D = initial ring deflection before vacuum is applied, H γ h uB ϕ K = = = = = = ft = soil cover, 102 pcf = unit weight of soil, — no water table, — no hydrostatic pressure at B, 25o = soil friction angle at the spring lines, 2.464 = ratio of horizontal to vertical effective soil stresses at soil slip Substituting values into Equation 10.13, p = 11.9 psi = vacuum at collapse Increasing ϕ by compacting the soil greatly increases the vacuum at collapse carefully placed and compacted so that ring deflection will not be excessive Ring Compression: Ring compression stress must be less than shortterm yield Long-term yield strength does not apply because, under constant deflection in select embedment, the plastic relaxes faster than the yield strength regresses See Chapter 20 Ring Inversion: Example In evaluating critical pres sure, P, it is assumed that: Specifications for the penstock of Example limit ring deflection to 5% Had ring deflection been 5%, what would be the internal vacuum at collapse? From Equation 10.13, p = 18.7 psi Because the maximum vacuum (atmospheric pressure) is only 14.7 psi, the pipe would not collapse This illustrates the importance of limiting ring deflection by soil compaction if the pipe is to be subjected to vacuum The basic deformation is from circle to ellipse Friction between soil and pipe is negligible There is no vacuum in the pipe and no water table Equation 10.13 applies p = 0; uB = 0; and, for high fills , σ y ≅ P A ≅ P The critical pressure P is the following in three different forms: Example P = 8Ed/(DR-1)3(1-K/rr) Solve Example if ring stiffness is neglected; i.e the pipe is so flexible that ring stiffness cannot be depended upon to support any of the soil load With 8% ring deflection, what is the vacuum at collapse? From Equation 10.13, neglecting the ring stiffness term, Ed/m3, vacuum at collapse is, p = 8.0 psi Ring stiffness does provide resistance to inversion in loose soil P = Ed/m3(1-K/rr) STABILITY DESIGN AND ANALYSIS Flexible Pipes Under High Fills With No Internal Vacuum and No Water Table One example of flexible pipes under high fills is drain pipes under sanitary landfills Some sanitary landfills reach heights of hundreds of feet They require drainage by a system of pipes The leachate may be so corrosive that plastic pipes are specified For plastic pipes, the embedment must be select, and ©2000 CRC Press LLC P = 1.7854 (F/ )d/(1-K/rr) (10.14) where K = (1+sinϕ)/(1-sinϕ), and r r = (1+d)3/(1-d)3 From Equations 10.14, it is clear that if K > rr, there is no soil slip regardless of soil pressure P In fact, P becomes negative Equation 10.14 is plotted in Figure 10-8 The vertical scale is dimensionless critical soil pressure term, P/(EI/r3) The soil friction angle of the sidefill is ϕ The horizontal scale is ring deflection d If the critical soil pressure term, P/(EI/r3), and the ring deflection term, d, locate a point below or to the left of a soil friction curve (ϕ-curve), the buried pipe is stable If the point falls above and to the right of a ϕ-curve, collapse is incipient — not imminent but possible Collapse may progress over a period of time, due to soil dynamics such as earth tremors, wetting and drying, etc Compaction of the embedment has a significant effect on stability No safety factor is included, but soil arching action assures a margin of safety If ring deflection is more than 20%, Equation 10.14 (Figure 10-8) loses accuracy because of nonelliptical ring deformation and ring stresses beyond elastic limit Collapse is not a problem if ring deflection is less than about 10% even for very loose granular soil (ϕ = 15o) In fact, pipes with more than 10% ring deflection are usually rejected for reasons other than structural instability Including a safety factor, if ring deflection is more than 10%, minimum soil friction angle should be increased — say to ϕ = 30° pipe stiffness is F/ = 234 psi for E = 400,000 psi From Equation 10.14 if P/(F/ ) = 600(75)/234(144) = 1.34, and ϕ = 15o, ring deflection at incipient collapse is d = 11.4% Ring deflection can be controlled by the quality and density of the sidefill From laboratory tests, select crushed stone compacted to 95% density AASHTO T99 (70% relative density) will hold ring deflection to less than d = 5% under 600 feet of cover at unit weight of 75 pcf With Soil Support — No Water Table or Vacuum The conditions for stability are assured if the sidefills are good granular soil, carefully compacted, and if the ring deflection is less than 10% Under conditions where mitigation is sought, live loads and height of soil cover can be limited Performance limit is soil slip of the sidefill At soil slip, the pressure of the pipe against the soil at spring line is equal to passive soil resistance Example P is soil pressure at the top of the pipe rr = (1+d)3/(1-d)3 From Equation 10.15, ring deflection, d, can be found at soil slip PVC piping is proposed for drainage under a sanitary landfill that is to be 600 ft high Unit weight of the landfill is 75 pounds per cubic feet Fifteen years are anticipated to complete the landfill The piping is to serve for 100 years What dimension ratio (DR) is required? DR is the ratio of outside pipe diameter to wall thickness Assume a 15-year yield strength of 5000 psi for PVC Safety factor is to be 1.5 From Equation 10.9, ring compression stress is, σ = 0.5γ H(1+d)DR For a long-term sanitary landfill, it is prudent, for cleaning the pipes, to hold ring deflection to nearly zero by compacted select sidefill Solving Equation 10.14 with d = 0, and with a safety factor of 1.5; DR = 21.3 Specify PVC pipe SDR 21(200) ASTM D2241 SDR is "standard dimension ratio." It is defined the same as DR; i.e., SDR = OD/t PVC pipes are resistant to corrosive leachate What is the maximum allowable ring deflection if embedment is loose with soil friction angle ϕ = 15°? From the Uni-Bell Handbook, for SDR = 21, the ©2000 CRC Press LLC Prr = Kσ y (10.15) Example What is the ring deflection of a flexible steel pipe at sidefill soil slip if D = 72 inches and H = ft? Embedment is poor, granular, loose soil Unit weight is γ = 100 pcf The soil friction angle is assumed to be ϕ = 15o from which, K = (1+sinϕ)/(1-sinϕ) = 1.7 P = γ H = 400 psf σy = γ Z = (100pcf)(4ft+3ft) = 700 psf For the first trial, let Z = ft Substituting these values into Equation 10.15, Prr = (400lb/ft2)(1+d)3/(1-d)3 = 1189 lb/ft2 Solving, d = 18% For a second trial, let Z = 4ft+2.5ft = 6.5 ft to account for the 18% reduction in vertical diameter of the deflected ring This solution yields d = 17% The above analysis is conservative because ring stiffness is ignored Ring stiffness is included in Figure 10-8 which comprises graphs at soil slip of soil pressure as a function of ring deflection and sidefill soil friction angle It is assumed that soil Figure 10-8 UNSATURATED SOIL — Soil pressure term at soil slip as a function of ring deflection and soil friction angle Empty steel pipes — no internal vacuum When used for design, a safety factor should be considered Liquefied Soil — Soil can liquefy if it is saturated, and shaken, and if density is less than about 80%, AASHTO T-180 The concept of liquefaction is as follows Pour loose dry sand into a quart jar to the top Carefully fill to the top with water Replace the lid Shake the jar, remove the lid and turn the jar upside down Liquefied soil gushes out because the sand is "shaken down." Repeat the experiment, but, this time, densify (tamp) the sand in layers as it is placed in the quart jar Then fill to the top with water, replace the lid, shake, remove the lid, and turn upside down The wet sand "hangs up" in the jar It has not liquefied In fact, the soil strength has increased because the sand is "shaken up." Figure 10-9 Conditions for collapse of a flexible ring in liquefied embedment ©2000 CRC Press LLC cover is high enough that H is essentially equal to Z Figure 10-9 shows interrelationships of the pertinent variables Two important conclusions are: Compaction of the embedment has a significant effect on pressure, P, at soil slip Soil does not slip if ring deflection is less than about 10% Therefore, maximum allowable ring deflection is often limited by specification, to 5%, including safety factor — or less if other performance limits prevail Example If height of cover is H = 12 ft, what is the ring deflection of a steel pipe at soil slip? D = 72 inches and t = 0.245 in The embedment is poor, loose soil for which γ = 100 pcf, and friction angle is φ = 15o EI/r3 = 0.78 psi P = 1200 psf = 8.33 psi The pressure term is P/(EI/r3) = 10.7 From Figure 10-9, d = 11% No problem is anticipated if ring deflection is less than 5% — even in this poor soil Flexible Pipes in Liquefied Soil Embedment If the embedment liquefies when a circular pipe is empty, the ring may be subjected to the hydrostatic pressures shown in Figure 10-8 If flotation is prevented, catastrophic collapse occurs from the bottom according to the classical equation, Pr3/EI = 3; or h = (E/4γ )(t/r)3 for plain pipe Example What is the height, h, of water table above the bottom of a steel pipe in embedment so loose that it can liquefy and cause catastrophic ring collapse? Pipe: D = 51 inches, t = 0.219, r/t = 117 Soil: γ = 125 pcf, saturated, ©2000 CRC Press LLC h P = height of water table above invert, = hγ Solving, h = 5.4 ft This illustrates the importance of densifying embedment soil — including soil under the haunches — if a water table could rise in the embedment With Soil Support and Internal Vacuum — No Water Table The performance limit for internal vacuum and/or external soil pressure is ring inversion Embedment usually prevents total collapse Critical vacuum, p, is sensitive to radius of curvature Ring deflection reduces critical vacuum Because vertical radius of curvature, ry , is greater than r; ring stiffness, EI/ry 3, is less than EI/r3, and the vacuum at collapse is less for a deflected ring than for a circular ring The stability analysis can include internal vacuum, p, and the resistance of ring stiffness which, for a plain pipe, is Ed/m3 The horizontal stresses on the infinitesimal cube, B, of Figure 10-6 can be equated to passive soil resistance (soil slip) Solving for vacuum, p, at soil slip, _ p(rr-1) = Kσ y - (PA-Ed/m3)rr UNSATURATED SOIL (10.16) For notation, see the more general form, Equation 10.17 Figure 10-10 shows graphs of Equation 10.16 for a plain steel pipe with D/t = 288, and ring deflection d = 10%, in granular embedment with two feet of cover It is noteworthy that critical vacuum is increased significantly by compacting the embedment (increased soil friction angle, φ) The effect of soil unit weight on critical vacuum is small Example A plain steel pipe is 51 inches in diameter with a 0.187 inch thick wall D/t = 274 D/t = 288 Figure 10-10 UNSATURATED SOIL — Example of graphs constructed for the design and analysis of the following buried steel pipe: D = 48 inches t = 0.167 inches H = ft γ = 100 pcf = soil unit weight ©2000 CRC Press LLC The height of soil cover is ft The soil is silty sand (SM) with soil friction angle φ = 25o (light compaction), and unit weight of about 100 pcf If the buried ring deflection is discovered to be d = 10%, what is the internal vacuum at soil slip? Because D/t is close to 288, Figure 10-10 may be used p = psi Had the soil been compacted such that φ = 35o, all else unchanged, the pipe could have withstood a vacuum of 12 psi And had the ring deflection been only 5% in compacted soil, the pipe could have withstood a vacuum of 26 psi, which is above atmospheric pressure By ring compression analysis, the critical vacuum would be increased tenfold For the design of pipes to withstand internal vacuum, a safety factor of 1.5 is recommended It is prudent to require that embedment soil be denser than critical Critical density can be evaluated in the soils laboratory Even without a water table, percolating water and earth tremors tend to shake loose soil down such that ring deflection could increase and reduce internal vacuum at collapse With Soil Support — With Water Table Above the Pipe: If the water table is above the top of the pipe, the soil is in no danger of liquefaction if density of the embedment is 90% Standard Proctor (ASTM D698 or AASHTO T-99) The height of water table, h, above ground surface, adds to the internal vacuum The worst case is an empty pipe with the water table above ground surface (flood level) See Figure 10-11 Critical vacuum includes water table above the pipe and effective soil pressure Using the stability analysis of Figure 10-6, but including ring stiffness and vacuum and water table, the equation of stability is, _ p(rr-1) = Kσ y + uB - (PA + πrγ w/2 - Ed/m3)rr SATURATED SOIL (10.17) where: p = vacuum and/or pressure due to flood level h above the pipe, ©2000 CRC Press LLC _ σy PA K φ uB h γw E d D m r t rr = = = = = = = = = = = = = = effective vertical soil stress at B, total vertical pressure at A, (1+sinφ)/(1-sinφ), soil friction angle, water pressure at B = (h+H+r)γ w, height of water table above ground surface, unit weight of water = 62.4 pcf, modulus of elasticity of steel = 30(10-6)psi, ring deflection (ellipse) = ∆ /D, circular diameter of the pipe, r/t = ring flexibility, D/2 = radius of the circular pipe, wall thickness, ry /rx The term, (πrγ w /2), is uplift pressure equivalent to buoyancy of the empty pipe If the pipe is full of water, this term is dropped from Equation 10.17 Noteworthy from Figure 10-11: A water table reduces the critical vacuum The effect of D/t on p is minor for values of D/t greater than 240 Soil becomes the primary resistance to vacuum The pipe is a lining The significant variables are ring deflection and soil density Example A steel pipe of diameter D = 51 inches and wall thickness t = 0.187 inch is buried under a soil cover of H = ft The embedment is loose granular soil with saturated unit weight of 125 pcf and φ = 15o The water table is at ground surface Ring deflection happens to be 10% What is the internal vacuum at ring collapse? Substituting values into Equation 10.17, the critical vacuum is p' = 0.4 psi, which leaves little margin of safety against collapse in a flood If ring deflection had been held to 5%, even in this poor soil, the vacuum at collapse would have been 3.8 psi which is equivalent to a flood 4.8 ft above ground surface In the example above, if embedment had been compacted such D/ t = 240 ©2000 CRC Press LLC that saturated unit weight was 130 pcf and φ = 35o, vacuum at collapse would be P' = 6.9 psi If ring deflection were 5% in this compacted embedment, "vacuum" at collapse would exceed 20 psi ring deflection at which collapse occurs? Specific gravity of soil grains is 2.65 (6.1%) REFERENCES 10-6 In Problem 10-4, what pipe stiffness, F/∆, would be needed to prevent collapse with a safety factor of w.r.t vacuum if ring deflection is 5% and vacuum is 11.4 psi? (24 psi) Murphy, C.E and Langner, C.G., (1985) Ultimate Pipe Strength Under Bending," Proceedings of ASME 4th International Offshore and Arctic Engineering Symposium, New York, N.Y PROBLEMS 10-1 What does Equation 10.3 reduce to if = 0, and ring compression stress term, σy /m, is negligible? (P'r3/EI = 3) 10-2 Derive Equation 10.6 for flexural stress: σf = (Et/D)(3do-3d)/(1-2do-2d) 10-3 A 10-inch PVC pipe Schedule 80 is to serve as a buried conduit for telephone cables under a river Suppose that for some reason the backfill is washed off the pipe at one location What is the head of water at which the pipe will collapse? From the Uni-Bell Handbook, DR = 18.13, and F/∆ = 370 psi (380 ft) 10-4 For a steel pipe penstock full of water, D = 96 inches in diameter , t = 0.375 inch, m = 128 = r/t, E = 30(106) psi, The pipe is buried in an embedment of silt and fine sand The height of soil cover is 16 ft At certain times during the year, the water table rises to ft below the ground surface The pipe stiffness is F/∆ = 5.5 psi The dry unit weight of soil is 100 pcf Soil friction angle is 15o The maximum vacuum that can occur is 11.4 psi What is the elliptical ©2000 CRC Press LLC 10-5 In Problem 10-4 what is the internal vacuum at collapse if the ring deflection is 3%? (39 psi) 10-7 What is the allowable external hydrostatic pressure plus internal vacuum on 150 class PVC pipe of nominal 10 inch diameter? (818 psi) E = 400 ksi Assumptions: σf = ksi Temperature, 40ο F ID = 9.82 in Safety factor = OD = 11.12 in No soil restraint (Soil liquefies) 10-8 If the pipe of Problem 10-7 is 10% out-ofround (ellipse), what is the allowable external hydrostatic pressure? (88 psi) 10-9 What is the external water head at collapse of a 4D PVC pipe, SDR 26, if a sudden internal vacuum of 12 psi occurs at the same time as the external water head? D/t = 25 E = 400 ksi (90.5 ft) 10-10 What is the allowable external water head on a 20D polyethylene pipe, DR 32.5, if the pipe is unburied and has an initial ovality (ring deflection) of 5%? From manufacturer's engineering data, short term E = 115 ksi 10-11 What horizontal soil-bearing capacity is required for a 36D PVC pipe, SDR 41? Pipe stiffness is F/∆ = 28 psi? Initial ring deflection due to careless installation is 15.9% and the vertical soil load on top of the pipe is P = ksf The soil is dry Safety factor is 1.0 at the point of collapse  ... stiffness of the ring resists inversion Soil supports the ring by holding it in a stable (near circular) shape Soil resists inversion of the ring Two basic modes of ring instability are: ring compression,... spring lines by methods of Chapter ©2000 CRC Press LLC x x Figure 10-6 SATURATED SOIL — Free-body-diagram of an infinitesimal soil cube at B, showing the stresses acting on it at incipient soil. .. of soil at point B, at soil slip, is soil passive resistance, σx = Kσ y Pressure Against Soil at Spring Lines _where σ = horizontal effective soil stress at B, _x σ y = vertical effective soil